The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper, we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving average and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian limit.
Accepté le :
DOI : 10.1051/ps/2019008
Mots-clés : Limit theorems, Lévy processes, moving averages, quadratic forms
@article{PS_2019__23__803_0,
author = {Nielsen, Mikkel Slot and Pedersen, Jan},
title = {Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages},
journal = {ESAIM: Probability and Statistics},
pages = {803--822},
publisher = {EDP-Sciences},
volume = {23},
year = {2019},
doi = {10.1051/ps/2019008},
mrnumber = {4045543},
zbl = {1506.60035},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2019008/}
}
TY - JOUR AU - Nielsen, Mikkel Slot AU - Pedersen, Jan TI - Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages JO - ESAIM: Probability and Statistics PY - 2019 SP - 803 EP - 822 VL - 23 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2019008/ DO - 10.1051/ps/2019008 LA - en ID - PS_2019__23__803_0 ER -
%0 Journal Article %A Nielsen, Mikkel Slot %A Pedersen, Jan %T Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages %J ESAIM: Probability and Statistics %D 2019 %P 803-822 %V 23 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2019008/ %R 10.1051/ps/2019008 %G en %F PS_2019__23__803_0
Nielsen, Mikkel Slot; Pedersen, Jan. Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 803-822. doi : 10.1051/ps/2019008. http://archive.numdam.org/articles/10.1051/ps/2019008/
[1] , On bilinear forms in Gaussian random variables and Toeplitz matrices. Probab. Theory Relat. Fields 79 (1988) 37–45. | DOI | MR | Zbl
[2] , and , Limit theorems for quadratic forms of Lévy-driven continuous-time linear processes. Stoch. Process. Their Appl. 126 (2016) 1036–1065. | DOI | MR | Zbl
[3] , , and , A continuous-time framework for ARMA processes. Preprint (2018). | arXiv
[4] , , and , Long-Memory Processes. Springer, Berlin (2016). | Zbl
[5] and , On the sample autocovariance of a Lévy driven moving average process when sampled at a renewal sequence. J. Stat. Plan. Inference 203 (2019) 20–38. | DOI | MR | Zbl
[6] , Lévy-driven CARMA processes. Ann. Inst. Stat. Math. 53 (2001) 113–124. | DOI | MR | Zbl
[7] and , Time Series: Theory and Methods. Springer Science & Business Media, NY (2013). | Zbl
[8] , and , Estimation for non-negative Lévy-driven CARMA processes. J. Bus. Econ. Stat. 29 (2011) 250–259. | DOI | MR | Zbl
[9] and , Existence and uniqueness of stationary Lévy-driven CARMA processes. Stoch. Process. Their Appl. 119 (2009) 2660–2681. | DOI | MR | Zbl
[10] and , A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process. J. Stat. Plan. Inference 143 (2013) 1295–1306. | DOI | MR | Zbl
[11] , and , Theory and Applications of Long-Range Dependence. Birkhäuser Boston Inc., Boston, MA (2003). | MR | Zbl
[12] , and , Multiple Stratonovich integral and Hu-Meyer formula for Lévy processes. Ann. Probab. 38 (2010) 2136–2169. | DOI | MR | Zbl
[13] and , Noncentral limit theorems for quadratic forms in random variables having long-range dependence. Ann. Probab. 13 (1985) 428–446. | MR | Zbl
[14] and , Central limit theorems for quadratic forms in random variables having long-range dependence. Probab. Theory Relat. Fields 74 (1987) 213–240. | DOI | MR | Zbl
[15] , and , Large Sample Inference for Long Memory Processes. World Scientific Publishing Company, Singapore (2012). | DOI | MR | Zbl
[16] and , A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle’s estimate. Probab. Theory Relat. Fields 86 (1990) 87–104. | DOI | MR | Zbl
[17] and , On stationary solutions of delay differential equations driven by a Lévy process. Stoch. Process. Their Appl. 88 (2000) 195–211. | DOI | MR | Zbl
[18] , Time Series Analysis, Vol. 2. Princeton University Press, Princeton (1994). | DOI | MR | Zbl
[19] and , Statistical inference for discrete-time samples from affine stochastic delay differential equations. Bernoulli 19 (2013) 409–425. | DOI | MR | Zbl
[20] , Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12 (2006) 1099–1126. | DOI | MR | Zbl
[21] and , Multivariate CARMA processes. Stoch. Process. Their Appl. 117 (2007) 96–120. | DOI | MR | Zbl
[22] and , Long-Range Dependence and Self-Similarity, in Vol. 45. Cambridge University Press, Cambridge (2017). | DOI | MR
[23] and , Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82 (1989) 451–487. | DOI | MR | Zbl
[24] , Lévy Processes and Infinitely Divisible Distributions. Translated fromthe 1990 Japanese original, revised by the author. Vol. 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999). | MR | Zbl
[25] , Limit theorems for the sample autocovariance of a continuous-time moving average process with long memory. Preprint (2015). | arXiv
Cité par Sources :
